Suppose you is able to solve the problem for a fixed size of state-space, i.e. construct the transition matrix and compute the steady state vector. Then you may use
FindSequenceFunction to guess how elements of the stationary probability vector depend on N.
Here is a simplistic example:
tm[n_Integer /; n > 1] :=
SparseArray[{{i_, j_} /; j - i == 1 && j <= n :> 1/2, {i_, i_} /; i < n :> 1/2, {n, n} -> 1}, {n, n}];
In[150]:= Table[PDF[StationaryDistribution[DiscreteMarkovProcess[1, tm[n]]], n], {n, 2, 12}]
Out[150]= {1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}
--Sasha
